Parabolic and Hyperbolic Equations - Core Definitions

Parabolic and hyperbolic evolution equations model time-dependent phenomena with fundamentally different character: diffusive smoothing versus wave propagation. Understanding their distinct properties is essential for both theory and applications.

DefinitionParabolic Equations

A second-order evolution equation: $\frac{\partial u}{\partial t} + L u = f$ is parabolic if $L$ is an elliptic spatial operator. The canonical form is: $u_t - \sum_{ij}\partial_i(a^{ij}(x,t)\partial_j u) + \sum_i b^i\partial_i u + cu = f$

with uniform ellipticity: $\sum a^{ij}\xi_i\xi_j \geq \lambda|\xi|^2$ .

DefinitionHyperbolic Equations

A second-order evolution equation: $\frac{\partial^2 u}{\partial t^2} + Lu = f$ is hyperbolic if it admits real characteristics. The canonical form is: $u_{tt} - \sum_{ij}a^{ij}(x,t)\partial_{ij}u + \text{lower order} = f$ with $(a^{ij})$ positive definite.

Key Differences:

Parabolic: First-order in time, infinite propagation speed, smoothing effect
Hyperbolic: Second-order in time, finite propagation speed, preserves singularities

ExampleStandard Examples

Parabolic:

Heat equation: $u_t = k\Delta u$
Fokker-Planck: $u_t = \nabla \cdot (D\nabla u - \mathbf{v}u)$
Reaction-diffusion: $u_t = D\Delta u + f(u)$

Hyperbolic:

Wave equation: $u_{tt} = c^2\Delta u$
Telegraph equation: $u_{tt} + au_t = c^2\Delta u$
Klein-Gordon: $u_{tt} - c^2\Delta u + m^2u = 0$

DefinitionWell-Posedness

For parabolic equations, initial-boundary value problems require:

Initial data: $u(x,0) = u_0(x)$
Boundary data for all $t > 0$

For hyperbolic equations, Cauchy problems require:

Initial position: $u(x,0) = u_0(x)$
Initial velocity: $u_t(x,0) = u_1(x)$

Remark

The distinction reflects physics: diffusion (parabolic) has no inertia—only current state matters. Wave propagation (hyperbolic) has inertia—velocity is an independent degree of freedom requiring separate initial data.